Tree representations of non-symmetric group-valued proximities

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Let X be a finite set and let d be a function from X x X into an
arbitrary group Q. An example of such a function arises by taking a tree T
whose vertices include X, assigning two elements of Q to each edge of T ( one
for each orientation of the edge), and setting d(i,j) equal to the product of the
elements along the directed path from i to j. We characterize conditions when
an arbitrary function d can be represented in this way, and show how such
a representation may be explicitly constructed. We also describe the extent
to which the underlying tree and the edge weightings are unique in such a
representation. These results generalize a recent theorem involving undirected
edge assignments by an Abelian group. The non-Abelian bi-directed case is of
particular relevance to phylogeny reconstruction in molecular biology.